An antique machine with C(n , 3) switches capable of processing integers in the range 0...2^{N}-1 has just been discovered. Each switch is associated to a distinct integer in 0...2^{N}-1 with exactly three ones in its binary representation. By setting switches associated with number X_{0},X_{1}...X_{M-1} to on, any integer Y passing through the machine will render a result of Y⊕X_{0}⊕X_{1}⊕...⊕X_{M-1}(here “⊕” stands for bitwise-XOR).

Further inspections reveal that contrary to what we assumed in problem B, some of the switches on the machine are damaged due to their old age. We are interested in whether a configuration transforming integer S into T still exists, and if so, the minimum number of switches that have to be set to on to make it possible.

WARNING: a naive algorithm might not be sufficient to solve this problem.

Further inspections reveal that contrary to what we assumed in problem B, some of the switches on the machine are damaged due to their old age. We are interested in whether a configuration transforming integer S into T still exists, and if so, the minimum number of switches that have to be set to on to make it possible.

WARNING: a naive algorithm might not be sufficient to solve this problem.

There are multiple test cases in the input file.

Each test case starts with two integers, N and M ( 1 ≤ N ≤ 20), representing the number of bits and the number of functioning switches, respectively. Two integers, S and T ( 0 ≤ S,T < 2^{N}), come in the next line, followed by another M lines, the ith one describing the value Vi associated to the ith switch ( 0 ≤ V_{i} < 2^{N}).

Two successive test cases are separated by a blank line. A case with N = 0 and M = 0 indicates the end of the input file, and should not be processed by your program.

Each test case starts with two integers, N and M ( 1 ≤ N ≤ 20), representing the number of bits and the number of functioning switches, respectively. Two integers, S and T ( 0 ≤ S,T < 2

Two successive test cases are separated by a blank line. A case with N = 0 and M = 0 indicates the end of the input file, and should not be processed by your program.

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